At a fixed parameter value t, assign each entire transformed clause polynomial (a factor) to one group g. Its value must depend only on the local assignment x_g and other groups' measurements. Changing an outside assignment while preserving its measurements must not change the factor. The paper ensures this by making each replacement input's outside dependence a linear combination of those measurements. Multiplying factors assigned to g gives a group weight A_g; use 1 if there are none. For an outside group {a,b} summarized by a+b, dependence only on a+b is allowed. Dependence on a alone generally is not: (0,1) and (1,0) share the summary but differ in a.

The condition that lets shared measurements separate factors

For group g and measurement list y_g, let X_g(y_g) contain exactly the local Boolean assignments producing y_g. This is a measurement fiber. With measurement a+b, X(0)={(0,0)}, X(1)={(0,1),(1,0)}, and X(2)={(1,1)}; X(3) is empty. Fix a measurement list for every group. Since groups use disjoint variables, choosing one assignment from each fiber produces one full assignment uniquely. Every full assignment belongs to one measurement tuple. With no measurements, all local assignments share the empty list. A group with no variables has one empty assignment, whose measurement rows, if any, all give zero.

AI Agents

A measurement of a variable group is an integer-weighted sum of its bits. For {a,b}, y=a+b gives 0, 1, or 2; (0,1) and (1,0) share value 1. A group can have several measurements, written C_g x_g, where g labels the group. The coefficient table C_g with rows (1,1) and (1,-1) gives the list (a+b,a-b). All factors use the same measurements for a given group. Choose a nonnegative integer D at least as large as each row's sum of absolute coefficient values. Every measurement lies between -D and D. This finite range can include unattainable values. A summary need not identify the original assignment.

Shared linear measurements of Boolean variable groups

OpenAI. The Exponential Time Hypothesis is False. Manuscript dated October 17, 2026, revised October 25, 2026. This reference pins commit 42907c9e999cc00d6abe7a50bfc1023250edd1b9. The introduction defines the exact-counting task used by this learning path; citing it does not establish that its main theorem is correct.

https://github.com/openai/math/blob/42907c9e999cc00d6abe7a50bfc1023250edd1b9/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf

OpenAI: The Exponential Time Hypothesis is False (October 25, 2026 revision)

A matrix C is a table of coefficients; C zeta is the list of its row measurements. Its kernel consists of difference vectors xi satisfying C xi=0. Two vectors have the same measurements exactly when their difference is in this kernel. A linear output B zeta is therefore determined by C zeta whenever B xi=0 for every xi in the kernel. For example, C zeta=zeta_1+zeta_2 determines 3zeta_1+3zeta_2, but not zeta_1. In the construction, choose a linear map M that fixes the kernel: M xi=xi there. Then B=I-M vanishes on the kernel, so the correction (I-M)zeta is determined by the measurements. I denotes the identity map.

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